An alternative to Witt vectors

The ring of Witt vectors associated to a ring R is a classical tool in algebra. We introduce a ring C(R) which is more easily constructed and which is isomorphic to the ring of Witt vectors W(R) for a perfect F_p-algebra R. It is obtained as the completion of the monoid ring ZR, for the multiplicative monoid R, with respect to the powers of the kernel of the natural map from ZR to R.Comment: slightly expanded introduction, proposition 4 added which gives a simple description of C(R) as an additive grou

Witt Vector Rings and the Relative de Rham Witt Complex

In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring $R$ and it is an important fact that the resulting objects are independent of all choices. The approach via presentations also sheds new light on our previous description of the ring of $p$-typical Witt vectors of a perfect $\mathbb{F}_p$-algebra as a completion of a semigroup algebra. We develop this description in different directions. For example, we show that the semigroup algebra can be replaced by any free presentation of $R$ equipped with a linear lift of the Frobenius automorphism. Using the result in the appendix by Umberto Zannier we also extend the description of the Witt vector ring as a completion to all $\bar{\mathbb{F}}_p$-algebras with injective Frobenius map.Comment: Appendix by Umberto Zannier; added the construction of a functorial noncommutative Witt vector ring with Frobenius Verschiebung and Teichm\"uller maps for noncommutative rings extending the commutative theor

Reduced operator algebras of trace-preserving quantum automorphism groups

Let $B$ be a finite dimensional C$^\ast$-algebra equipped with its canonical trace induced by the regular representation of $B$ on itself. In this paper, we study various properties of the trace-preserving quantum automorphism group \G of $B$. We prove that the discrete dual quantum group \hG has the property of rapid decay, the reduced von Neumann algebra L^\infty(\G) has the Haagerup property and is solid, and that L^\infty(\G) is (in most cases) a prime type II$_1$-factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced $C^\ast$-algebra C_r(\G), and the existence of a multiplier-bounded approximate identity for the convolution algebra L^1(\G).Comment: Section 6 removed and replaced by a more general solidity resul

Locally C∗-equivalent algebras

AbstractIt is shown that a Banach star algebra is a C∗-algebra in an equivalent norm, if each of its commutative closed star subalgebras is a C∗-algebra in an equivalent norm. This theorem has several interesting consequences