Ilan Hirshberg

Ilan Hirshberg
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Ilan Hirshberg

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Mathematics - Operator Algebras (20)
Mathematics - Dynamical Systems (2)
Mathematics - Logic (2)
Mathematics - K-Theory and Homology (1)

Publications Authored By Ilan Hirshberg

We show that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O_2, or of the CAR algebra. Read More

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Read More

We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. Read More

We show that if X is a finite dimensional locally compact Hausdorff space, then the crossed product of C_0(X) by any automorphism has finite nuclear dimension. This generalizes previous results, in which the automorphism was required to be free. As an application, we show that group C*-algebras of certain non-nilpotent groups have finite nuclear dimension. Read More

We show that the Calkin algebra is not countably homogeneous, in the sense of continuous model theory. We furthermore show that the connected component of the unitary group of the Calkin algebra is not countably homogeneous. Read More

This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on C*-algebras, introduced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This definition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and Z-absorption. Read More

In this paper we define a Rokhlin property for automorphisms of non-unital C*-algebras and for endomorphisms. We show that the crossed product of a C*-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C*-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Read More

We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C*-algebras, where Z denotes the Jiang-Su algebra. Read More

We study a tracial notion of Z-absorption for simple, unital C*-algebras. We show that if A is a C*-algebra for which this property holds then A has almost unperforated Cuntz semigroup, and if in addition A is nuclear and separable we show this property is equivalent to having A being Z-absorbing. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property. Read More

We show that nuclear C*-algebras have a refined version of the completely positive approximation property, in which the maps that approximately factorize through finite dimensional algebras are convex combinations of order zero maps. We use this to show that a separable nuclear C*-algebra A which is closely contained in a C*-algebra B embeds into B. Read More

We consider families of E_0-semigroups continuously parametrized by a compact Hausdorff space, which are cocycle-equivalent to a given E_0-semigroup \beta. When the gauge group of $\beta$ is a Lie group, we establish a correspondence between such families and principal bundles whose structure group is the gauge group of \beta. Read More

We exhibit a unital simple nuclear non-type-I C*-algebra into which the Jiang-Su algebra does not embed unitally. This answers a question of M. R{\o}rdam. Read More

We study permanence properties of the classes of stable and so-called D-stable C*-algebras, respectively. More precisely, we show that a C_0(X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for C*-algebras absorbing the Jiang--Su algebra Z tensorially). Furthermore, we prove that if D is a K_1-injective strongly self-absorbing C*-algebra, then A absorbs D tensorially if and only if all its fibres do, again provided that X is finite-dimensional. Read More

Let A be a unital separable C*-algebra, and D a K_1-injective strongly self-absorbing C*-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility. Read More

Let E be a C*-correspondence over a C*-algebra \A with non-degenerate faithful left action. We show that E admits sufficiently many essential representations (i.e. Read More

We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. Read More

We consider a class of C*-algebras associated to one parameter continuous tensor product systems of Hilbert modules, which can be viewed as continuous counterparts of Pimsner's Toeplitz algebras. By exhibiting a homotopy of quasihomomorphisms, we prove that those algebras are $K$-contractible. One special case is closely related to the Rieffel-Wiener-Hopf extension of a crossed product by R considered by Rieffel and by Pimsner and Voiculescu, and can be used to produce a new proof of Connes' analogue of the Thom isomorphism and in particular of Bott periodicity. Read More

Let \alpha:G --> G be an endomorphism of a discrete amenable group such that [G:\alpha(G)]Read More

We classify spectrum-preserving endomorphisms of stable continuous-trace C^*-algebras up to inner automorphism by a surjective multiplicative invariant taking values in finite dimensional vector bundles over the spectrum. Specializing to automorphisms, this gives a different approach to results of Phillips and Raeburn. Read More