We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of `peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is weak* closed.
We study projections in the corona algebra of $C(X)\otimes K$ where $X=[0,1],[0,\infty),(-\infty,\infty)$, and $[0,1]/\{0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. Then we also characterize the conditions for two projections to be equal in $K_0$-group, Murray-von Neumann equivalent, unitarily equivalent, and homotopic from the weakest to the strongest. In light of these characterizations, we construct examples showing that any two equivalence notions do not coincide, which s…
We give an explicit formula for the time projection in an arbitrary von Neumann algebra from which all its basic properties can be easily derived. The analysis of the situation when this time projection is a conditional expectation is also performed.
Generalizing von Neumann's result on type II$_1$ von Neumann algebras, we characterize lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, we give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.
Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform. (Namely, as two circle algebras of the same dimension, that are swapped by the Fourier transform, plus a few points.) The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of Theta functions.
We characterize the $C^\star$-algebras for which openness of projections in their second duals is preserved under Murray-von Neumann equivalence. They are precisely the extensions of the annihilator $C^\star$-algebras by the commutative $C^\star$-algebras. We also show that the annihilator $C^\star$-algebras are precisely the $C^\star$-algebras for which all projections in their second duals are open.
Let $(P_1,...,P_n)$ be an $n$--tuple of projections in a unital $C^*$--algebra $å$. We say $\pn$ is complete in $å$ if $å$ is the linear direct sum of the closed subspaces $P_1å,...,P_nå$. In this paper, we give some necessary and sufficient conditions for the completeness of $\pn$ and discuss the perturbation problem and topology of the set of all complete $n$--tuple of projections in $å$. Some interesting and significant results are obtained in this paper.
In this paper we present new structural information about the multiplier algebra Mult (A) of a sigma-unital purely infinite simple C*-algebra A, by characterizing the positive elements a in Mult(A) that are strict sums of projections belonging to A. If a is not in A and is not a projection, then the necessary and sufficient condition for a to be a strict sum of projections belonging to A is that the norm ||a||>1 and that the essential norm ||a||_ess >=1. Based on a generalization of the Perera-Rordam weak divisibility of separable simple C*-algebras of real rank zero to all sigma-unital simple…
The closed one-sided ideals of a C*-algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C*-algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B** which also lies in the weak* closure of A. Although this seems quite natural, the proof requires a set of new techniques which may may be viewed as a noncommutative version o…
In [BEZ] the notion of a complete one-sided M-ideal for an operator space X was introduced as a generalization of Alfsen and Effros' notion of an M-ideal for a Banach space [AE72]. In particular, various equivalent formulations of complete one-sided M-projections were given. In this paper, some sharper equivalent formulations are given in the special situation that $X = \mathcal{A}$, a $C^*$-algebra (in which case the complete left M-projections are simply left multiplication on $\mathcal{A}$ by a fixed orthogonal projection in $\mathcal{A}$ or its multiplier algebra). The proof of the first e…
We introduce the notion of the partial group algebra with projections and relations and show that this C*-algebra is a partial crossed product. Examples of partial group algebras with projections and relations are the Cuntz-Krieger algebras and the unitization of C*-algebras of directed graphs.
In this paper we investigate whether positive elements in the multiplier algebras of certain finite C*-algebras can be written as finite linear combinations of projections with positive coefficients (PCP). Our focus is on the category of underlying C*-algebras that are separable, simple, with real rank zero, stable rank one, finitely many extreme traces, and strict comparison of projections by the traces. We prove that the strict comparison of projections holds also in the multiplier algebra of the stabilizer algebra. Based on this result and under the additional hypothesis that the multiplier…
We first prove that in a sigma-finite von Neumann factor M, a positive element $a$ with properly infinite range projection R_a is a linear combination of projections with positive coefficients if and only if the essential norm ||a||_e with respect to the closed two-sided ideal J(M) generated by the finite projections of M does not vanish. Then we show that if ||a||_e>1, then a is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra. Secondly, we provide a necessary condition for a positive operator …
The universal C*-algebra generated by n projections has been described. As an immediate corollary one obtains structure theorem for a pair of projections and the solution to an associated index problem. This puts the study of a pair of projections in the proper perspective of noncommutative geometry.
We give an overview of the question: which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive elements can be written as a sum of finitely many projections. We focus on von Neumann algebras, on purely infinite simple C*-algebras, and on their associated multiplier algebras.
Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $π:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $π(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal…
We give a complete description of which unital graph C*-algebras are semiprojective, and use it to disprove two conjectures by Blackadar. To do so, we perform a detailed analysis of which projections are properly infinite in such C*-algebras.
The manuscript is devoted to the construction of $Ψ^\star$- algebras containing the Bergman projection on the unit ball $B_n$ of $\mathbb C^n$. We consider the $C^\star$-algebra $\mathcal{L}(L^2(B_n))$ of bounded operator acting on the Hilbert space of square integrable functions on $B_n$ with respect to the standard probability measure. We search for spectral invariant Fréchet subalgebras of $\mathcal{L}(L^2(B_n))$ containing the Bergman projection $P$ which is defined on $L^2(B_n)$ with values in the space of holomorphic functions. We use the commutator method as introduced by Gramsch. This …
Let $X$ be a unit interval or a unit circle and let $B$ be a $σ_p$-unital, purely infinite, simple $C\sp*$-algebra such that its multiplier algebra $M(B)$ has real rank zero. Then we determine necessary and sufficient conditions for a projection in the corona algebra of $C(X)\otimes B$ to be liftable to a projection in the multiplier algebra. This generalizes a result proved by L. Brown and the author \cite{BL}. The main technical tools are divided into two parts. The first part is borrowed from the author's previous paper(JFA 260 (2011)). The second part is a proposition showing that we can p…
In this paper we consider certain proejctions in the corona algebra of $C(X)\otimes B$ associated to $(p_0, p_1, \dots, p_n)$ where $p_i: X_i \to \mt_s$ a continuous projection valued section to the multiplier algebra of a stable $C\sp*$-algebra $B$ for each i. Here $X_i$'s are closed intervals given by a partition $\{x_1, \dots, x_n\}$ on the interior of $X$ and adjacent sections differ by compacts at each partition point. Assuming a kind of homogeneity on the projection we characterize when two such projections are homotopy equivalent.
Let A be a C*-algebra and A** its enveloping von Neumann algebra. C. Akemann suggested a kind of non-commutative topology in which certain projections in A** play the role of open sets. The adjectives "open", "closed", "compact", and "relatively compact" all can be applied to projections in A**. Two operator inequalities were used by Akemann in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in A**, but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, …
Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]\sqsubseteq [q]$ iff for every primitive ideal $\mathfrak p$ of $A$ either $p/\mathfrak p\preceq q/\mathfrak p\preceq (1- q)/\mathfrak p$ or $p/\mathfrak p\succeq q/\mathfrak p \succeq (1-q)/\mathfrak p.$ We prove that $p$ is central iff $[p]$ is $\sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparab…
Broadly speaking, this paper is concerned with dual spaces of operator algebras. More precisely, we investigate the existence of what we call Lebesgue projections: central projections in the bidual of an operator algebra that detect the weak-$*$ continuous part of the dual space. Associated to any such projection is a Lebesgue decomposition of the dual space. We are particularly interested in Lebesgue projections in the context of inclusions of operator algebras. We show how their presence is intimately connected with an extension property for the inclusion reminiscent of a classical theorem o…
It is shown that a strongly self-absorbing C*-algebra is of real rank zero and absorbs the Jiang-Su algebra if it contains a nontrivial projection. We also consider cases where the UCT is automatic for strongly self-absorbing C*-algebras, and K-theoretical ways of characterizing when Kirchberg algebras are strongly self-absorbing.
In this paper we study a relationship between systems of $n$ subspaces and representations of $*$-algebras generated by projections. We prove that irreducible nonequivalent $*$-representations of $*$-algebras $\mathcal P_{4,com}$ generate all nonisomorphic transitive quadruples of subspaces of a finite dimensional space.
It is shown that for every measure $m$ on projections in a $W^*$-algebra of type $I_2$, there exists a Hilbert-valued orthogonal vector measure $μ$ such that $\|μ(p)\|^2= m(p)$ for every projection $p$. With regard to J. Hamhalter's result (Proc. Amer. Math. Soc., 110 (1990), 803-806) it means that the assertion is valid for an arbitrary $W^*$-algebra.
Let ${\cal A}$ be a von Neumann algebra and ${\cal P}_{\cal A}$ the manifold of projections in ${\cal A}$. There is a natural linear connection in ${\cal P}_{\cal A}$, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of $\mathbb{C}^n$. In this paper we show that two projections $p,q$ can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ${\cal A}$), if and only if $$ p\wedge q^\perp\sim p^\perp\wedge q, $$ where $\sim$ stands for the Murray-von Neumann equivalence of projections. …
We study the universal C^*-algebras generated by n projections $p_1, >..., p_n$ subject to the relation $p_1+... p_n = λ1$, $λ\in \mathbb R$. The questions of when these C^*-algebras are type I, nuclear or exact are considered. It is proved also that among these algebras there is continuum of mutually nonisomorphic ones.
We introduce the Rohlin property and the approximate representability for finite group actions on stably projectionless C*-algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra and show some classification results of these actions. This study is based on the work of Izumi, Robert's classification theorem and Kirchberg's central sequence C*-algebras.
We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which a…
Assume that A is a purely infinite simple C*-algebra whose K_0 is a torsion group, namely, contains no free element. Then a positive element a in A can be written as a finite sum of projections in A if and only if either a is a projection or the norm ||a||>1.
We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan $^*$-isomorphisms. In particular, we prove that two von Neumann algebras without type I$_1$ direct summands are Jordan $^*$-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. Šemrl, which is a generalization of Wigner's theorem.
We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated by Ortega, Rordam and Thiel. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary sub-C*-algebra of A.
Let $\mathbb{G}$ be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in $L^\infty(\mathbb{G})$ preserved by the scaling group and idempotent states on the dual quantum group $\widehat{\mathbb{G}}$. As a byproduct we give a simple proof that normal integrable coideals in $L^\infty(\mathbb{G})$ which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of $\mathbb{G}$.
In this short note, we prove that for a $C^*$-algebra $å$ generated by $n$ elements, $M_{k}(\tildeå)$ is generated by $k$ mutually unitarily equivalent and almost mutually orthogonal projections for any $k\ge \de(n)=\min\big\{k\in\mathbb N\,|\,(k-1)(k-2)\ge 2n\big\}$. Then combining this result with recent works of Nagisa, Thiel and Winter on the generators of $C^*$--algebras, we show that for a $C^*$-algebra $å$ generated by finite number of elements, there is $d\ge 3$ such that $M_d(\tilde A)$ is generated by three mutually unitarily equivalent and almost mutually orthogonal projections. Fur…
An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite. Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras. Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-alg…
We investigate a set of functional equations defining a projection in the noncommutative 2-torus algebra $A_θ$. The exact solutions of these provide various generalisations of the Powers-Rieffel projection. By identifying the corresponding $K_0(A_θ)$ classes we get an insight into the structure of projections in $A_θ$.
We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the …
In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using $\KK$-theory and prove a result which asserts that there is a unitary of the form `identity + compact' which gives the unitary equivalence of two projections if the `essential codimension' of two projections vanishes for certain $C\sp*$-algebras employing the proper asymptotic unitary equivalence of $\KK$-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of $C(X)\otimes B$ where $X$ is $[0,1]$, $(-\infty, \infty)$, $[0,\infty)$, …
We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the can…
The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker versions of the operator norm localization property (ONL) which turn out to characterize those technical geometric conditions. We use this to obtain new rigidity results for nonmetrizable coarse spaces. As an application, we provide a novel partial answer to a question of White and Willett about Cartan subalgebras of uniform Roe algebras. We also study embeddi…
We investigate the set-theoretic properties of the lattice of projections in the Calkin algebra of a separable infinite-dimensional Hilbert space in relation to those of the Boolean algebra $P(ω)/{\rm fin}$, which is isomorphic to the sublattice of diagonal projections. In particular, we prove some basic consistency results about the possible cofinalities of well-ordered sequences of projections and the possible cardinalities of sets of mutually orthogonal projections that are analogous to well-known results about $P(ω)/{\rm fin}$.
The Gelfand - Naĭmark theorem supplies the one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants can be defined by algebraical methods. This article contains a pure algebraical construction of (noncommutative) covering projections with finite cyclic groups of covering transformations.
We construct a simple, nuclear, stably projectionless C*-algebra W which has trivial K-theory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self-absorbing C*-algebras as an analogue of the Cuntz algebra O_2. In this context, we show that every nondegenerate endomorphism of W is approximately inner and we construct a trace-preserving embedding of W into the central sequences algebra M(W)_\infty \cap W'. We conjecture that W\otimes W is isomorphic to W and we note some implications of this, for example that W would be absorbed tenso…
The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.
The Gelfand - Naĭmark theorem supplies the one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants can be defined by algebraical methods. This article contains a pure algebraical construction of inverse limits in the category of (noncommutative) covering projections. It is proven that Moyal planes are inverse limits of covering projections of noncommutative tori.
The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions. We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these results is the co-duality for coideals in algebras of functions on locally compact quantum groups and the concept of a normal coideal.
Every invariant linear manifold for a CSL-algebra is a closed subspace if, and only if, each non-zero projection in the projection lattice is generated by finitely many atoms. In the case of a nest, this condition is equivalent to the condition that every non-zero projection in the nest has an immediate predecessor (the nest of orthogonal complements is well ordered). The invariant linear manifolds of a nest algebra are totally ordered by inclusion if, and only if, every non-zero projection in the nest has an immediate predecessor.
Gelfand - Naĭmark theorem supplies a one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space. Generalizations of several topological invariants may be defined by algebraic methods. For example Serre Swan theorem states that complex topological $K$-theory coincides with $K$-theory of $C^*$-algebras. This article devoted to the noncommutative generalization of infinite covering projections. Infinite covering projections of spectral triples are also discussed. …
We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$. We also define and study a natural generalization of the Fell topology to arbitrary Banach space representations of a locally compact group. We give several applications in terms of stability of rigidity under perturbations. Among them, we show a Banach-space version of the Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent for $σ$-compact locally compact groups. Another kind of applications…
For each $α\in (0,1)$, $A_α$ denotes the universal $C^*$-algebra generated by two unitaries $u$ and $v$, which satisfy the commutation relation $uv=\exp (2πiα)vu$. We consider the order four automorphism $σ$ of $A_α$ defined by $σ(u)=v$, $σ(v)=u^{-1}$ and describe a method for constructing projections in the fixed point algebra $A_α^σ$, using Rieffel's imprimitivity bimodules and Jacobi's theta functions. In the case $α=q^{-1}$, $q\in {\mathbf Z}$, $q\geq 2$, we give explicit formulae for such projections and find a lower bound for the norm of the Harper operator $u+u^* +v+v^*$.
We develop some tools for manipulating and constructing projections in C*-algebras. These are then applied to give short proofs of some standard projection homotopy results, as well as strengthen some fundamental classical results for C*-algebras of real rank zero, specifically on liftings, excising pure states and Kadison's transitivity theorem. Lastly, we investigate some order properties of the set of projections in C*-algebras of real rank zero, building on our previous work.
Assume that A is a unital separable simple C*-algebra with real rank zero, stable rank one, strict comparison of projections, and that its tracial simplex T(A) has a finite number of extremal points. We prove that every self-adjoint element a in A in the kernel of all tracial states is the sum of two commutators in A and that every positive element of A is a linear combination of projections with positive coefficients. Assume that A is as above but σ-unital. Then an element (resp. a positive element) a of A is a linear combination (resp. a linear combination with positive coefficients) of proj…
We show that a map between projection lattices of semi-finite von Neumann algebras can be extended to a Jordan $*$-homomorphism between the von Neumann algebras if this map is defined in terms of the support projections of images (under the linear map) of projections and the images of orthogonal projections have orthogonal support projections. This has numerous fundamental applications in the study of isometries and composition operators on quantum symmetric spaces and is of independent interest, since it provides a partial generalization of Dye's Theorem without the requirement that the initi…
